#6629403

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628896
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,19,22,15)(12,16,24,29)(13,17,25,30)(20,31,34,26)(21,32,35,27)(23,36,37,28)(33,38,40,39);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,26)(17,27)(18,19)(23,33)(24,34)(25,35)(28,38)(29,31)(30,32)(36,39)(37,40);(9,12)(10,16)(11,20)(13,23)(14,24)(15,26)(17,28)(18,29)(19,31)(21,33)(22,34)(25,37)(27,38)(30,36)(32,39)(35,40);(9,13)(10,17)(11,21)(12,23)(14,25)(15,27)(16,28)(18,30)(19,32)(20,33)(22,35)(24,37)(26,38)(29,36)(31,39)(34,40);(1,3,4,7)(2,6,5,8)(11,22)(15,19)(20,34)(21,35)(26,31)(27,32)(33,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,e^2=fi,b^a=bf,c^a=cg,d^a=dh,e^b=ei,h^c=hi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341043, #72341151, #72341158
maximal normal subgroup of	#72341043, #72341151, #72341158

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536599);